Proof, series from k=1 to n of cos(2*pi*k/(2n+1)) equals -1/2
I have worked out a proof for relationships of complex numbers, and part of my proof requires me to prove this...
-1/2 = series from k=1 to n of cos(2*pi*k/(2n+1))
that is, for any positive integer n
I've wrote a program that evaluated this for an extended amount of integers n, in every case the series was equal to -1/2
However, I need a formal proof for this problem.
Another proof that I have to solve is this
0 = series from k=1 to n-1 of cos(pi*k/n)
that is, for any positive integer n greater than 1
this is easier to deduce, given that the values in the series will cancel each other out, or else be equivalent to 0
that being said, I'm looking for a formal proof for this series to